Derive one-dimensional heat equation

Setting. A thin homogeneous bar of uniform cross-section A has its lateral surface insulated, so heat flows only along its length (the x-axis). Let u(x, t) be the temperature; ρ = density, σ = specific heat, K = thermal conductivity (all constants).

Law of conduction. The rate of heat flow across a cross-section in the +x direction is −K A ∂u/∂x. (The minus sign shows that heat flows from hot to cold.)

Element. Consider the slice between x and x + δx.

  heat in →  ┌────────────┐  → heat out
   at x      │  mass      │    at x+δx
             │  ρ A δx    │
             └────────────┘

• Heat entering at x per unit time: Q₁ = −K A (∂u/∂x)ₓ
• Heat leaving at x + δx per unit time: Q₂ = −K A (∂u/∂x)_{x+δx}

Net heat gained per unit time:

Q₁ − Q₂ = K A [ (∂u/∂x)_{x+δx} − (∂u/∂x)ₓ ]

Heat absorbed. To warm a mass ρ A δx at the rate ∂u/∂t requires

σ ρ A δx ∂u/∂t

Equating:

σρ A δx ∂u/∂t = K A [ (∂u/∂x)_{x+δx} − (∂u/∂x)ₓ ]

∂u/∂t = (K/σρ) [ (∂u/∂x)_{x+δx} − (∂u/∂x)ₓ ] / δx

As δx → 0, the bracket divided by δx tends to ∂²u/∂x². Writing c² = K/(σρ) (the diffusivity of the material):

∂u/∂t = c² ∂²u/∂x²

This is the one-dimensional heat (diffusion) equation.